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I have discovered that current mathematics education often teaches students to emphasize all polynomial equations should have complex solutions.

But starting from non-polynomial equations (e.g. $\sin x=x$), even in fact there are infinitely many complex solutions, the current mathematics education often teaches students to ignore them, even for the occasions on just needing the analyses of the number of the solutions. These attitudes make me upset as I for example feel that they have forgotten most transcendental functions are also defined in $\mathbb{C}$ and they are not respect the world of complex numbers.

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    $\begingroup$ It's not entirely clear what you're asking for. Do you want ways to deal with this problem? Or do you want a historical explanation? Something else? $\endgroup$ – adamblan Mar 15 '14 at 21:52
  • $\begingroup$ @adamblan: e.g. study the reasons of the particular phenomenon appearing in the current mathematics education. $\endgroup$ – doraemonpaul Mar 16 '14 at 1:11
  • $\begingroup$ Why somebody want to close this question? Does Mathematics Educators SE really becomes the site which only welcome the mathematics educators' views on mathematics education but not welcome the students' views on mathematics education? $\endgroup$ – doraemonpaul Mar 16 '14 at 18:46
  • $\begingroup$ @doraemonpaul I'm sorry this question was closed. Personally, my understanding is that we don't teach this because complex numbers simply make things more complex. At least, for the most part. But complex numbers are often unavoidable in polynomials. Often, it depends on the math course you are in. If it is a course that can avoid complex numbers all together, then it makes it easier for the students and the teachers. I have, in fact, found that I understand complex analysis a lot better than my teachers (high school). $\endgroup$ – Simply Beautiful Art Feb 3 '16 at 23:19
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Rather than explaining why complex solutions to transcendental equations are not discussed, I think it is worth explaining why the complex solutions to polynomial equations are - that, I think, comes down to the Fundamental Theorem of Algebra. The heavy emphasis on complex roots of polynomials could help serve as a precursor for the result, so that when students finally are introduced to it, they're not quite as surprised.

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I think the answer to your question is that techniques for solving transcendental equations for complex variables are fairly complicated (with the exception, perhaps, of Newton's method), so they tend not to be taught in elementary courses.

In particular, analogs of the bisection method for functions of a complex variable include the Lehmer-Schur algorithm and Wilf's algorithm, which uses the argument principle from complex analysis. This might be mentioned in a very good undergraduate complex analysis course, but otherwise it probably doesn't come up until graduate courses on complex analysis or numerical methods.

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  • $\begingroup$ If I am be the teacher, facing for solving e.g. $\sin x=x$ , I will teach my student that e.g. $x=a+bi$ , where $a$ and $b$ are the real solutions of $\begin{cases}a=\sin a\cosh b\\b=\cos a\sinh b\end{cases}$ , the trivial solution $x=0$ is the special case of $b=0$ . Plotting the complex plane of the system of equations will clearly show the pattern of the solutions of $\sin x=x$ . Showing them on textbooks are really difficult? $\endgroup$ – doraemonpaul Mar 15 '14 at 23:30
  • $\begingroup$ Unfortunately, ignoring the analyses of complex number solutions on (systems) of non-polynomial equations becomes likely the endemic spreading over the mathematics world. $\endgroup$ – doraemonpaul Mar 15 '14 at 23:30
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    $\begingroup$ @doraemonpaul I think you're overestimating the importance of complex numbers. In the vast majority of applications, only real solutions are helpful. $\endgroup$ – Jim Belk Mar 15 '14 at 23:52
  • $\begingroup$ But the analyses of complex number solutions are really mentioned quite well for the polynomial equations (although for the systems of polynomial equations are tends to be excepted). The most disappointing point is that e.g. it appears the gap of values of the complex number solutions between the polynomial equations and the non-polynomial equations. Especially for pure mathematics, concepts are taking more series roles. $\endgroup$ – doraemonpaul Mar 16 '14 at 0:24
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    $\begingroup$ I'm honestly not sure why one would care about such solutions. $\endgroup$ – user37 Mar 16 '14 at 3:28

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